3.228 \(\int (c e+d e x)^m (a+b \cosh ^{-1}(c+d x))^2 \, dx\)

Optimal. Leaf size=206 \[ -\frac{2 b^2 (e (c+d x))^{m+3} \text{HypergeometricPFQ}\left (\left \{1,\frac{m}{2}+\frac{3}{2},\frac{m}{2}+\frac{3}{2}\right \},\left \{\frac{m}{2}+2,\frac{m}{2}+\frac{5}{2}\right \},(c+d x)^2\right )}{d e^3 (m+1) (m+2) (m+3)}-\frac{2 b \sqrt{-c-d x+1} (e (c+d x))^{m+2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+2}{2},\frac{m+4}{2},(c+d x)^2\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^2 (m+1) (m+2) \sqrt{c+d x-1}}+\frac{(e (c+d x))^{m+1} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e (m+1)} \]

[Out]

((e*(c + d*x))^(1 + m)*(a + b*ArcCosh[c + d*x])^2)/(d*e*(1 + m)) - (2*b*Sqrt[1 - c - d*x]*(e*(c + d*x))^(2 + m
)*(a + b*ArcCosh[c + d*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, (c + d*x)^2])/(d*e^2*(1 + m)*(2 + m)*S
qrt[-1 + c + d*x]) - (2*b^2*(e*(c + d*x))^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 +
m/2}, (c + d*x)^2])/(d*e^3*(1 + m)*(2 + m)*(3 + m))

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Rubi [A]  time = 0.318661, antiderivative size = 218, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {5866, 5662, 5763} \[ -\frac{2 b^2 (e (c+d x))^{m+3} \, _3F_2\left (1,\frac{m}{2}+\frac{3}{2},\frac{m}{2}+\frac{3}{2};\frac{m}{2}+2,\frac{m}{2}+\frac{5}{2};(c+d x)^2\right )}{d e^3 (m+1) (m+2) (m+3)}-\frac{2 b \sqrt{1-(c+d x)^2} (e (c+d x))^{m+2} \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};(c+d x)^2\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^2 (m+1) (m+2) \sqrt{c+d x-1} \sqrt{c+d x+1}}+\frac{(e (c+d x))^{m+1} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^m*(a + b*ArcCosh[c + d*x])^2,x]

[Out]

((e*(c + d*x))^(1 + m)*(a + b*ArcCosh[c + d*x])^2)/(d*e*(1 + m)) - (2*b*(e*(c + d*x))^(2 + m)*Sqrt[1 - (c + d*
x)^2]*(a + b*ArcCosh[c + d*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, (c + d*x)^2])/(d*e^2*(1 + m)*(2 +
m)*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]) - (2*b^2*(e*(c + d*x))^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 +
m/2}, {2 + m/2, 5/2 + m/2}, (c + d*x)^2])/(d*e^3*(1 + m)*(2 + m)*(3 + m))

Rule 5866

Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
]

Rule 5662

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcC
osh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5763

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_
)]), x_Symbol] :> Simp[((f*x)^(m + 1)*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2,
(3 + m)/2, c^2*x^2])/(f*(m + 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), x] + Simp[(b*c*(f*x)^(m + 2)*Hypergeometric
PFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/(Sqrt[-(d1*d2)]*f^2*(m + 1)*(m + 2)), x] /; FreeQ[{
a, b, c, d1, e1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[d1, 0] && LtQ[d2, 0] &&  !
IntegerQ[m]

Rubi steps

\begin{align*} \int (c e+d e x)^m \left (a+b \cosh ^{-1}(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int (e x)^m \left (a+b \cosh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac{(e (c+d x))^{1+m} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e (1+m)}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{(e x)^{1+m} \left (a+b \cosh ^{-1}(x)\right )}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{d e (1+m)}\\ &=\frac{(e (c+d x))^{1+m} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e (1+m)}-\frac{2 b (e (c+d x))^{2+m} \sqrt{1-(c+d x)^2} \left (a+b \cosh ^{-1}(c+d x)\right ) \, _2F_1\left (\frac{1}{2},\frac{2+m}{2};\frac{4+m}{2};(c+d x)^2\right )}{d e^2 (1+m) (2+m) \sqrt{-1+c+d x} \sqrt{1+c+d x}}-\frac{2 b^2 (e (c+d x))^{3+m} \, _3F_2\left (1,\frac{3}{2}+\frac{m}{2},\frac{3}{2}+\frac{m}{2};2+\frac{m}{2},\frac{5}{2}+\frac{m}{2};(c+d x)^2\right )}{d e^3 (1+m) (2+m) (3+m)}\\ \end{align*}

Mathematica [A]  time = 0.429451, size = 178, normalized size = 0.86 \[ \frac{(c+d x) (e (c+d x))^m \left (\left (a+b \cosh ^{-1}(c+d x)\right )^2-\frac{2 b (c+d x) \left (\frac{b (c+d x) \text{HypergeometricPFQ}\left (\left \{1,\frac{m}{2}+\frac{3}{2},\frac{m}{2}+\frac{3}{2}\right \},\left \{\frac{m}{2}+2,\frac{m}{2}+\frac{5}{2}\right \},(c+d x)^2\right )}{m+3}+\frac{\sqrt{1-(c+d x)^2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+2}{2},\frac{m+4}{2},(c+d x)^2\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{\sqrt{c+d x-1} \sqrt{c+d x+1}}\right )}{m+2}\right )}{d (m+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)^m*(a + b*ArcCosh[c + d*x])^2,x]

[Out]

((c + d*x)*(e*(c + d*x))^m*((a + b*ArcCosh[c + d*x])^2 - (2*b*(c + d*x)*((Sqrt[1 - (c + d*x)^2]*(a + b*ArcCosh
[c + d*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, (c + d*x)^2])/(Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]) +
(b*(c + d*x)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, (c + d*x)^2])/(3 + m)))/(2 +
m)))/(d*(1 + m))

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Maple [F]  time = 2.108, size = 0, normalized size = 0. \begin{align*} \int \left ( dex+ce \right ) ^{m} \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^m*(a+b*arccosh(d*x+c))^2,x)

[Out]

int((d*e*x+c*e)^m*(a+b*arccosh(d*x+c))^2,x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^m*(a+b*arccosh(d*x+c))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b^{2} \operatorname{arcosh}\left (d x + c\right )^{2} + 2 \, a b \operatorname{arcosh}\left (d x + c\right ) + a^{2}\right )}{\left (d e x + c e\right )}^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^m*(a+b*arccosh(d*x+c))^2,x, algorithm="fricas")

[Out]

integral((b^2*arccosh(d*x + c)^2 + 2*a*b*arccosh(d*x + c) + a^2)*(d*e*x + c*e)^m, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e \left (c + d x\right )\right )^{m} \left (a + b \operatorname{acosh}{\left (c + d x \right )}\right )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)**m*(a+b*acosh(d*x+c))**2,x)

[Out]

Integral((e*(c + d*x))**m*(a + b*acosh(c + d*x))**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arcosh}\left (d x + c\right ) + a\right )}^{2}{\left (d e x + c e\right )}^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^m*(a+b*arccosh(d*x+c))^2,x, algorithm="giac")

[Out]

integrate((b*arccosh(d*x + c) + a)^2*(d*e*x + c*e)^m, x)